Part 1: Find the Eigenfunction Expansion Consider the function f (same as in the previous problem) defined on the interval [0, 10] as follows, 3 x, f(x)= x = [0,5], 5 3. x = [5, 10]. Find the coefficients C, of the eigenfunction expansion of function f, 00 f(x) = co+cn Yn(x), where yo=1 (which is not a unit eigenfunction), while yn, for n = 1, 2, 3, ., are unit the eigenfunctions of the Regular Sturm-Liouville system -y" = Ay. y' (0) = 0, y' (10) = 0. Note: The constant eigenfunction yo= 1 is not a unit eigenfunction, since yo yo = 10. Therefore, the formula for co is co= the eigenfunctions yn, for n = 1, 2, 3, ... to be unit eigenfunctions. Therefore, the formula for the coefficients c, is C = fyn- f-yo We choose yo yo Σ Co= Σ Cn =

Let x (t) x' (t) x' (t) If x (0) x₁(t) = = = = = sin( x ₂ (t) 2 = sin( x1(t) x2(t) ] Put the eigenvalues in ascending order when you enter x₁(t), x₂(t) below. -17x₁(t) 8 x ₁ (t) 1 -3 [:] sin( t) + sin( t) + t) + cos( be a solution to the system of differential equations: and x'(0) t) + + cos( 16 x ₂ (t) 2 7x₂(t) = -28 16 cos ( t) + t) cos( t) + t) " find x(t).

If a > 0, find the eigenvalues of x" + λax = 0, x (0) = x(b) = 0.

Let x(t) x₁ (t) x' (t) x₁ (t): = = If x (0) = x₂(t): = = = ] Put the eigenvalues in ascending order when you enter x₁ (t), x₂(t) below. exp( t) + exp( x₁(t) x₂(t) -27 x₁(t) + 12x₂(t) -56 x₁(t) + 25 x₂(t) 4 be a solution to the system of differential equations: -2 exp( find x(t). t) + exp( t) t)